THE CALCULUS OF LOGIC

GEORGE BOOLE

Cambridge and Dublin Mathematical Journal
Vol. III (1848), pp. 183-98

In a work lately published[1], I have exhibited the application of a new and peculiar form of Mathematics to the expression of the operations of the mind in reasoning. In the present essay I design to offer such an account of a portion of this treatise as may furnish a correct view of the nature of the system developed. I shall endeavour to state distinctly those positions in which its characteristic distinctions consist, and shall offer a more particular illustration of some features which are less prominently displayed in the (p. 184)[2] original work. The part of the system to which I shall confine my observations is that which treats of categorical propositions, and the positions which, under this limitation, I design to illustrate, are the following:

(1) That the business of Logic is with the relations of classes, and with the modes in which the mind contemplates those relations.

(2) That antecedently to our recognition of the existence of propositions, there are laws to which the conception of a class is subject,—laws which are dependent upon the constitution of the intellect, and which determine the character and form of the reasoning process.

(3) That those laws are capable of mathematical expression, and that they thus constitute the basis of an interpretable calculus.

(4) That those laws are, furthermore, such, that all equations which are formed in subjection to them, even though expressed under functional signs, admit of perfect solution, so that every problem in logic can be solved by reference to a general theorem.

(5) That the forms under which propositions are actually exhibited, in accordance with the principles of this calculus, are analogous with those of a philosophical language.

(6) That although the symbols of the calculus do not depend for their interpretation upon the idea of quantity, they nevertheless, in their particular application to syllogism, conduct us to the quantitative conditions of inference.

It is specially of the two last of these positions that I here desire to offer illustration, they having been but partially exemplified in the work referred to. Other points will, however, be made the subjects of incidental discussion. It will be necessary to premise the following notation.

The universe of conceivable objects is represented by 1 or unity. This I assume as the primary and subject conception. All subordinate conceptions of class are understood to be formed from it by limitation, according to the following scheme.

Suppose that we have the conception of any group of objects consisting of

s, and others, and that

, which we shall call an elective symbol, represents the mental operation of selecting from that group all the

s which it contains, or of fixing the attention upon the

s to the exclusion of all which are not

the mental operation of selecting the

s, and so on; then, 1 or the universe being the subject conception, we shall have

and so on.

In like manner we shall have

Furthermore, from consideration of the nature of the mental operation involved, it will appear that the following laws are satisfied.

Representing by

any elective symbols whatever,

From the first of these it is seen that elective symbols are distributive in their operation; from the second that they are commutative. The third I have termed the index law; it is peculiar to elective symbols.

The truth of these laws does not at all depend upon the nature, or the number, or the mutual relations, of the individuals included in the different classes. There may be but one individual in a class, or there may be a thousand. There may be individuals common to different classes, or the classes may be mutually exclusive. All elective symbols are distributive, and commutative, and all elective symbols satisfy the law expressed by (3).

These laws are in fact embodied in every spoken or written language. The equivalence of the expressions "good wise man" and "wise good man," is not a mere truism, but an assertion of the law of commutation exhibited in (2). And there are similar illustrations of the other laws.

With these laws there is connected a general axiom. We have seen that algebraic operations performed with elective symbols represent mental processes. Thus the connexion of two symbols by the sign + represents the aggregation of two classes into a single class, the connexion of two symbols

as in multiplication, represents the mental operation of selecting from a class

those members which belong also to another class

, and so on. By such operations the conception of a class is modified. But beside this the mind has the power of perceiving relations of equality among classes. The axiom in question, then, is that if a relation of equality is perceived between two classes, that relation remains unaffected when both subjects are equally modified by the operations above described. (A). This axiom, and not "Aristotle's dictum," is the real foundation of all reasoning, the form and character of the process being, however, determined by the three laws already stated.

It is not only true that every elective symbol representing a class satisfies the index law (3), but it may be rigorously demonstrated that any combination of elective symbols

(

..), which satisfies the law

(

..)ⁿ =

(

..), represents an intelligible conception,—a group or class defined by a greater or less number of properties and consisting of a greater or less number of parts.

The four categorical propositions upon which the doctrine of ordinary syllogism is founded, are

All Ys are Xs.	A,
No Ys are Xs.	E,
Some Ys are Xs.	I,
Some Ys are not Xs.	O.

We shall consider these with reference to the classes among which relation is expressed.

A. The expression All

s represents the class

and will therefore be expressed by

, the copula are by the sign =, the indefinite term,

s, is equivalent to Some

s. It is a convention of language, that the word Some is expressed in the subject, but not in the predicate of a proposition. The term Some

s will be expressed by

, in which

is an elective symbol appropriate to a class

, some members of which are

s, but which is in other respects arbitrary. Thus the proposition

will be expressed by the equation

E. In the proposition, No

s are

s, the negative particle appears to be attached to the subject instead of to the predicate to which it manifestly belongs.[3] We do not intend to say that those things which are not-

s are

s, but that things which are

s are not-

s. Now the class not-

s is expressed by 1 -

; hence the proposition No

s are

s, or rather All

s are not-

s, will be expressed by

I. In the proposition Some

s are

s, or Some

s are Some

s, we might regard the Some in the subject and the Some in the predicate as having reference to the same arbitrary class

, and so write

but it is less of an assumption to refrain from doing this. Thus we should write

' referring to another arbitrary class

'.

O. Similarly, the proposition Some

s are not-

s, will be expressed by the equation

It will be seen from the above that the forms under which the four categorical propositions A, E, I, O are exhibited in the notation of elective symbols are analogous with those of pure language, i.e. with the forms which human speech would assume, were its rules entirely constructed upon a scientific basis. In a vast majority of the propositions which can be conceived by the mind, the laws of expression have not been modified by usage, and the analogy becomes more apparent, e.g. the interpretation of the equation

is, the class

consists of all

s which are not-

s and of all

s which are not-Xs.

[1] The Mathematical Analysis of Logic, being an Essay towards a Calculus of Deductive Reasoning. Cambridge, MacMillan; London, G. Bell.

[2]The Mathematical Analysis of Logic

[3]There are two ways in which the proposition, No Xs are Ys, can be understood. 1st, In the sense of All Xs are not-Y, In the sense of It is not true that any Xs are Ys, i.e. the proposition "Some Xs are Ys". The former of these are categorical proposition. The latter is an assertion respecting a proposition, and its expression belongs to a distinct part of the elective system. It appears to me that it is the latter sense, which is really adopted by those who refer the negative, not, to the copula. To refer it to the predicate is not a useless refinement, but a necessary step, in order to make the proposition truly a relation between classes. I believe it will be found that this step is really taken in the attempts to demonstrate the Aristotelian rules of distribution.

The transposition of the negative is a very common feature of language. Habit renders us almost insensible to it in our own language, but when in another language the same principle is differently exhibited, as in the Greek, οὺ φημὶ for φημὶ οὺ, it claims attention.

General Theorems relating to Elective Functions.

We have now arrived at this step,—that we are in possession of a class of symbols

, &c. satisfying certain laws, and applicable to the rigorous expression of any categorical proposition whatever. It will be our next business to exhibit a few of the general theorems of the calculus which rest upon the basis of those laws, and these theorems we shall afterwards apply to the discussion of particular examples.

Of the general theorems I shall only exhibit two sets: those which relate to the development of functions, and those which relate to the solution of equations.

Theorems of Development.

(1) If

be any elective symbol, then

the coefficients

(1),

(0), which are quantitative or common algebraic functions, are called the moduli, and

and 1 -

the constituents.

(2) For a function of two elective symbols we have

in which

(11),

(10), &c. are quantitative, and are called the moduli, and

(1 -

), &c. the constituents.

(3) Functions of three symbols,

in which

(111),

(110), &c. are the moduli, and

(1 -

), &c. the constituents.

From these examples the general law of development is obvious. And I desire it to be noted that this law is a mere consequence of the primary laws which have been expressed in (1), (2), (3).

THEOREM. If we have any equation

(

..) = 0, and fully expand the first member, then every constituent whose modulus does not vanish may be equated to 0.

This enables us to interpret any equation by a general rule.

RULE. Bring all the terms to the first side, expand this in terms of all the elective symbols involved in it, and equate to 0 every constituent whose modulus does not vanish.

For the demonstration of these and many other results, I must refer to the original work. It must be noted that on p. 66[4], z has been, through mistake, substituted for

, and that the reference on p. 80[5] should be to Prop. 2.

As an example, let us take the equation

Here

(

) =

+ 2

- 3

, whence the values of the moduli are

so that the expansion (9) gives

which is in fact only another form of (11a). We have, then, by the Rule

the former implies that there are no Xs which are not-Ys, the latter that there are no Ys which are not-Xs, these together expressing the full significance of the original equation.

We can, however, often recombine the constituents with a gain of simplicity. In the present instance, subtracting (12) from (11b), we have

that is, the class

is identical with the class

. This proposition is equivalent to the two former ones.

All equations are thus of equal significance which give, on expansion, the same series of constituent equations, and all are interpretable.

[4]The Mathematical Analysis of Logic

[5]Ibid.

General Solution of Elective Equations.

(1) The general solution of the equation

(

) = 0, in which two elective symbols only are involved,

being the one whose value is sought, is

The coefficients

are here the moduli.

(2) The general solution of the equation

(

) = 0,

being the symbol whose value is to be determined, is

the coefficients of which we shall still term the moduli. The law of their formation will readily be seen, so that the general theorems which have been given for the solution of elective equations of two and three symbols, may be regarded as examples of a more general theorem applicable to all elective equations whatever. In applying these results it is to be observed, that if a modulus assume the form 0/0 it is to be replaced by an arbitrary elective symbol

, and that if a modulus assume any numerical value except 0 or 1, the constituent of which it is a factor must be separately equated to 0. Although these conditions are deduced solely from the laws to which the symbols are obedient, and without any reference to interpretation, they nevertheless render the solution of every equation interpretable in logic. To such formulae also every question upon the relations of classes may be referred. One or two very simple illustrations may suffice[6].

(1) Given